Pole Mapping with Optimal Zeros
We saw in the preceding sections that both the impulse-invariant and
the matched- transformations map poles from the left-half
plane
to the interior of the unit circle in the
plane via
where
![$ s_i$](http://www.dsprelated.com/josimages_new/pasp/img1736.png)
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
![$ s$](http://www.dsprelated.com/josimages_new/pasp/img144.png)
![$\vert\mbox{im\ensuremath{\left\{s\right\}}}\vert<\pi/T$](http://www.dsprelated.com/josimages_new/pasp/img1767.png)
![$ z$](http://www.dsprelated.com/josimages_new/pasp/img76.png)
Therefore, an obvious generalization is to map the poles according to
Eq.(8.8), but compute the zeros in some optimal way, such as by
Prony's method [449, p. 393],[273,297].
It is hard to do better Eq.(8.8) as a pole mapping from
to
, when aliasing is avoided, because it preserves both the resonance
frequency and bandwidth for a complex pole [449]. Therefore,
good practical modeling results can be obtained by optimizing the
zeros (residues) to achieve audio criteria given these fixed poles.
Alternatively, only the least-damped poles need be constrained in this
way, e.g., to fix and preserve the most important resonances of a
stringed-instrument body or acoustic space.
Next Section:
Modal Expansion
Previous Section:
Matched Z Transformation